Problem 40 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=-2 x^{3}-1$$

Step-by-Step Solution

Answer

Graph the cubic function, noting key points and end behavior.

Step 1: Identify the function type

The given function is a cubic polynomial, which generally has the form \( ax^3 + bx^2 + cx + d \). In this case, \( f(x) = -2x^3 - 1 \). Here, \( a = -2 \) and \( d = -1 \), with the coefficients of \( x^2 \) and \( x \) being zero.

Step 2: Determine the function's characteristics

Since \( a = -2 \) and it is negative, the cubic function has a downward opening with one local maximum and one local minimum. It will pass through the y-axis at the point \( (0, -1) \), as the constant term \( d = -1 \) represents the y-intercept.

Step 3: Find key points

To graph the function, calculate the value of \( f(x) \) for several \( x \) values. For example, \( x = 0 \) gives \( f(0) = -1 \), \( x = 1 \) gives \( f(1) = -3 \), \( x = -1 \) gives \( f(-1) = 1 \). These points will help plot the graph.

Step 4: Sketch the graph

With the computed points and understanding of the cubic function's behavior, draw the curve. Start from the left with an upward approach, pass through the calculated keys points such as \((-1, 1)\), \((0, -1)\), \((1, -3)\) and continue downward on the right.

Step 5: Analyze symmetry and end behavior

Check that the graph is symmetric about the origin, which it is not in this case, confirming it is an odd function overall. The end behavior for a negative cubic function will have the left side going to infinity, and the right side going to negative infinity. Ensure that these behaviors are depicted in your sketch.

Key Concepts

Cubic Polynomials

A cubic polynomial is a type of polynomial equation of the form \( ax^3 + bx^2 + cx + d \). In this form, \( a \), \( b \), \( c \), and \( d \) are constants, with the highest power of the variable being three. This makes it a degree three polynomial.
In our specific example, the function \( f(x) = -2x^3 - 1 \) is a simplified cubic polynomial. Here, the coefficients of \( x^2 \) and \( x \) are zero, which simplifies the expression greatly. The coefficient \( a = -2 \) dictates the nature of the curve in terms of its steepness and orientation. The constant \( d = -1 \) determines the y-intercept or where the graph will intersect the y-axis.
Since the coefficient for \( x^3 \) is negative, the graph of the polynomial will have a distinctive shape with a downward direction, much like an inverted "S".

Graphing Functions

Graphing functions involves plotting points derived from the function's equation on a coordinate plane to visually represent the behavior of the equation. Here's how you can graph \( f(x) = -2x^3 - 1 \):

• Start by identifying the y-intercept. For this function, it occurs at the point \((0, -1)\).
• Choose a range of \( x \) values both positive and negative, and calculate their corresponding \( f(x) \) values. For example, at \( x = 1 \), \( f(1) = -3 \) and at \( x = -1 \), \( f(-1) = 1 \).
• Plot these calculated points on the graph to give an outline of the function's path.
• Draw a smooth curve connecting these points, starting at the top left and curving downward to the right, respecting the general behavior indicated by the polynomial's terms.

When graphing cubic functions like this, expect one or two turns in the graph because the degree of the polynomial is three.

End Behavior Analysis

End behavior refers to where the graph of a polynomial heads as \( x \) approaches positive or negative infinity. This behavior is crucial in understanding how the graph behaves at its ends, regardless of the complexity of the middle region.
For a function like \( f(x) = -2x^3 - 1 \):

• As \( x \) approaches negative infinity, the term \(-2x^3\) dominates, meaning \( f(x) \) heads towards positive infinity. This indicates the graph rises.
• Conversely, as \( x \) approaches positive infinity, \( f(x) \) will decrease, trending strongly towards negative infinity. This denotes the graph falls.

This particular function's end behavior makes a mirrored image of a standard cubic curve, due to the negative \( ax^3 \) coefficient, confirming the downward opening or orientation.

Function Characteristics

Understanding function characteristics involves determining several important aspects of the function, such as intercepts, symmetry, and extrema points.

• Y-intercept: For \( f(x) = -2x^3 - 1 \), the y-intercept is at \((0, -1)\), indicating where the function crosses the y-axis.
• Symmetry: Check for symmetry by analyzing whether the function is even, odd, or neither. Here, the polynomial is not symmetrical around the origin or any axis but aligns with odd function behaviors.
• Maximum and Minimum: Given it's a cubic function, expect a local maximum and minimum but only one point each, making it simple yet significant to graph.

These characteristics help define the graph shape and aid in plotting, ensuring accurate portrayal of the polynomial's nature and behavior.